Are Borel sets in $\mathbb{R}^n$ Cartesian products of Borel sets in $\mathbb{R}$?

Question

It is well-known that a Cartesian product of Borel sets is a Borel set. Let $B(\mathbb{R})$ denote the Borel sigma algebra on $\mathbb{R}$ and $B(\mathbb{R}^n)$ denote the Bore sigma algebra on $\mathbb{R}^n$. Thus, $B(\mathbb{R})\times \dots \times B(\mathbb{R}) \subset B(\mathbb{R}^n)$.

But does the equality hold? I.e., is it true that any Borel set in $\mathbb{R}^n$ can be written as a Cartesian product of Borel sets of $\mathbb{R}$?

Answer 1

No. The easiest example would probably be $B(1) := \{x \in \mathbb{R}^n : \|x\| < 1\}$, which is in $B(\mathbb{R}^n)$ but is not the Cartesian product of any subsets of $\mathbb{R}$, Borel-measurable or otherwise.